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The Third British Mathematical Olympiad
1967年第三届英国奥林匹克数学竞赛
  1. a, b are the roots of x2 + Ax + 1 = 0, and c, d are the roots of x2 + Bx + 1 = 0. Prove that (a - c)(b - c)(a + d)(b + d) = B2 - A2.
  2. Graph x8 + xy + y8 = 0, showing stationary values and behaviour for large values. [Hint: put z = y/x.]
  3. (a) The triangle ABC has altitudes AP, BQ, CR and AB > BC. Prove that AB + CR ≥ BC + AP. When do we have equality ?
    (b) Prove that if the inscribed and circumscribed circles have the same centre, then the triangle is equilateral.
  4. We are given two distinct points A, B and a line l in the plane. Can we find points (in the plane) equidistant from A, B and l? How do we construct them ?
  5. Show that (x - sin x)(π - x - sin x) is increasing in the interval (0, π/2).
  6. Find all x in [0, 2π] for which 2 cos x ≤ |√(1 + sin 2x) - √(1 - sin 2x)| ≤ √2.
  7. Find all reals a, b, c, d such that abc + d = bcd + a = cda + b = dab + c = 2.
  8. For which positive integers n does 61 divide 5n - 4n ?
  9. None of the angles in the triangle ABC are zero. Find the greatest and least values of cos2A + cos2B + cos2C and the values of A, B, C for which they occur.
  10. A collects pre-1900 British stamps and foreign stamps. B collects post-1900 British stamps and foreign special issues. C collects pre-1900 foreign stamps and British special issues. D collects post-1900 foreign stamps and British special issues. What stamps are collected by (1) no one, (2) everyone, (3) A and D, but not B ?
  11. The streets for a rectangular grid. B is h blocks north and k blocks east of A. How many shortest paths are there from A to B ?
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